Volumetric imaging of holographic optical traps

ABSTRACT

A method and system for manipulating object using a three dimensional optical trap configuration. By use of selected hologram on optical strap can be configured as a preselected three dimensional configuration for a variety of complex uses. The system can include various optical train components, such as partially transmissive mirrors and Keplerian telescope components to provide advantageously three dimensional optical traps.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/974,716, filed Oct. 16, 2007, which claims priority from U.S.Provisional Application 60/852,252, filed Oct. 17, 2006, both of whichare incorporated herein by reference in their entirety.

This invention is directed toward volumetric imaging of holographicoptical traps. More particularly, the invention is directed to a methodand system for creating arbitrary pro-selected three-dimensional (3D)configurations of optical traps having individually specified opticalcharacteristics. Holographic techniques are used to modify individualtrap wavefronts to establish pre-selected 3D structures havingpredetermined properties and are positionable independently in threedimensional space to carry out a variety of commercially useful tasks.

The United States Government has certain rights in this inventionpursuant to a grant from the National Science Foundation through grantnumber DMR-0451589.

BACKGROUND OF THE INVENTION

There is, a well developed technology of using single light beams toform an optical trap which applies optical forces from the focused beamflight to confine an object to a particular location in space. Theseoptical traps, or optical tweezers, have enabled fine scale manipulationof objects for a variety of commercial purposes. In addition, linetraps, or extended optical tweezers, have been created which act as aone dimensional potential energy landscape for manipulating mesoscopicobjects. Such line traps can be used to rapidly screen interactionsbetween colloidal aid biological particles which find uses in biologicalresearch, medical diagnostics and drug discovery. However, theseapplications require methods of manipulation diagnostics and drugdiscovery. However, these applications require methods of manipulationfor projecting line traps with precisely defined characteristics whichprevent their use in situations with high performance demands. Further,the low degrees of freedom and facility of use for such line trapsreduces the ease of use and limits the types of uses available.

SUMMARY OF THE INVENTION

The facility and range of applications of optical traps is greatlyexpanded by the method and system of the invention in which 3D intensitydistributions are created by holography. These 3D representations arecreated by holographically translating optical traps through an opticaltrain's focal plane and acquiring a stack of two dimensional images inthe process. Shape phase holography is used to create a pre-selected 3Dintensity distribution which has substantial degrees of freedom tomanipulate any variety of object or mass for any task.

Various aspects of the invention are described hereinafter; and theseand other improvements are described in greater detail below, includingthe drawings described in the following section.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an optical train for performing a method of theinvention;

FIG. 2A illustrates a particular optical condition with z<0 for anobjective lens in the system of FIG. 1; FIG. 2B illustrates the opticalcondition for z=0 for the objective lens of FIG. 1 and FIG. 2Cillustrates the optical condition for z>0 for the objective lens of FIG.1;

FIG. 3A illustrates a 3D reconstruction of an optical tweezerpropagating along the z axis; FIG. 3B illustrates a cross-section ofFIG. 3A along an xy plane; FIG. 3C illustrates a cross-section of FIG.3A along a yz plane; FIG. 3D illustrates a cross-section of FIG. 3Aalong an xz plane; FIG. 3E illustrates a volumetric reconstruction of 35optical tweezers arranged in a body-centered cubic lattice of the typeshown in FIG. 3F;

FIG. 4A illustrates a 3D reconstruction of a cylindrical lens lineoptical tweezer; FIG. 4B illustrates a cross-section of FIG. 4A along anxy plane; FIG. 4C illustrates a cross-section of FIG. 4A along a yzplane; and FIG. 4D illustrates a cross-section of FIG. 4A along an xzplane; and

FIG. 5A illustrates a 3D reconstruction of a holographic optical trapfeaturing diffraction-limited convergence to a single focal plane; FIG.5B illustrates a cross-section of FIG. 5A along a xy plane; FIG. 5Cillustrates a cross-section of FIG. 5A along a yz plane; and FIG. 5Dillustrates a cross-section of FIG. 5A along an xz plane.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

An optical system for performing, methods of the invention isillustrated generally at 10 in FIG. 1. A beam of light 20 is output froma frequency-doubled solid-state laser 30, preferably a Coherent Verdisystem operating at a wavelength of λ=532 nm. The beam of light 20 isdirected to an input pupil 40 of a high-numerical-aperture objectivelens 50, preferably a Nikon 100×Plan Apo, NA 1.4, oil immersion systemthat focuses the beam of light 20 into an optical trap (not shown). Thebeam of light 20 is imprinted with a phase-only hologram by acomputer-addressed liquid-crystal spatial light modulator 60 (“SLM 60”),preferably a Hamamatsu X8267 PPM disposed in a plane conjugate to theobjective lens' 50 input plane. Computer 95 executes conventionalcomputer software to generate the appropriate hologram using the SLM 60.As a result, the light field, ψ(r), in the objective lens' 50 focalplane is related to the field ψ(ρ) in the plane of the SLM 60 by theFraunhofer transform,

$\begin{matrix}{{{\psi (r)} = {{- \frac{}{\lambda \; f}}{\int_{\Omega}^{\;}{{\psi (\rho)}\ {\exp \left( {{- }\frac{2\pi}{\lambda \; f}{r \cdot \rho}} \right)}{^{2}\rho}}}}},} & (1)\end{matrix}$

where ƒ is the objective's focal length, where Ω is the optical train'saperture, and where we have dropped irrelevant phase factors. Assumingthat the beam of light 20 illuminates the SLM 60 with a radiallysymmetric amplitude profile, u(ρ), and uniform phase, the field in theSLM's plane may be written as,

ψ(ρ)=u(ρ)exp(iφ(ρ)),  (2)

where φ(ρ) is the real-valued phase profile imprinted on the beam oflight 20 by the SUM 60. The SLM 60 in our preferred form of the system10 imposes phase shifts between 0 and 2 π radians at each pixel of a768×768 array. This two-dimensional phase array can be used to project acomputer-generated phase-only hologram, φ(ρ), designed to transform thesingle optical tweezer into any desired three-dimensional configurationof optical traps, each with individually specified intensities andwavefront properties.

Ordinarily, the pattern of holographic optical traps would be put to useby projecting it into a fluid-borne sample mounted in the objectivelens' 50 focal plane. To characterize the light field, we instead mounta front-surface mirror 70 in the sample plane. This mirror 70 reflectsthe trapping light back into the objective lens 50, which transmitsimages of the traps through the partially reflecting mirror 70 to acharge-coupled device (CCD) camera 80, preferably a NEC TI-324AII. Inour implementation, the objective lens 50, the camera 80 and cameraeyepiece (not shown), are mounted in a conventional optical Microscope(not shown) and which is preferably a Nikon TE-2000U.

Three-dimensional reconstructions of the optical traps' intensitydistribution can be obtained by translating the mirror 70 relative tothe objective lens 50. Equivalently, the traps can be translatedrelative to the mirror 70 by superimposing the parabolic phase function,

$\begin{matrix}{{{\phi_{z}(\rho)} = {- \frac{{\pi\rho}^{2}z}{\lambda \; f^{2}}}},} & (3)\end{matrix}$

onto the hologram φ₀(ρ) encoding a particular pattern of traps. Thecombined hologram, φ₀(ρ)=φ₀(ρ)+φ_(z)(ρ) mod 2 π, projects the samepattern of traps as φ₀(ρ) but with each trap translated by −z alongoptical axis 90 of the system 10. The resulting image obtained from thereflected light represents a cross-section of the original trappingintensity at distance z from the focal plane of the objective lens 50.Translating the traps under software control by computer 95 isparticularly convenient because it minimizes changes in the opticaltrain's properties due to mechanical motion and facilitates moreaccurate displacements along the optical axis 90. Images obtained ateach value of z are stacked up to yield a complete volumetricrepresentation of the intensity distribution.

As shown schematically in FIGS. 2A-2C, the objective lens 50 capturesessentially all of the reflected light for z<0. For z>0, however, theoutermost rays of the converging trap are cut off by the objective lens'50 output pupil 105, and the contrast is reduced accordingly. This couldbe corrected by multiplying the measured intensity field by a factorproportional to z for z>0. The appropriate factor, however, is difficultto determine accurately, so we present only unaltered results.

FIG. 3A shows a conventional optical tweezer 100 reconstructed in themanner described hereinbefore and displayed as an isointensity surfaceat 5 percent peak intensity and in three cross-sections (FIGS. 3B-3D).The representation in FIG. 3A is useful for showing the overallstructure of the converging light, and the cross-sections of FIGS. 3B-3Dprovide an impression of the three dimensional light field that willconfine an optically trapped object. The angle of convergence of 63° inimmersion oil obtained from these data is consistent with an overallnumerical aperture of 1.4. The radius of sharpest focus, r_(min)≈0.2 μm,is consistent with diffraction-limited focusing of the beam of light 20.

These results highlight two additional aspects of this reconstructiontechnique. The objective lens 50 is designed to correct for sphericalaberration when the beam of light 20 passing through water is refractedby a glass coverslip. Without this additional refraction, the projectedoptical trap 100 actually is degraded by roughly 20λ of sphericalaberration, introduced by the objective lens 50. This reduces theapparent numerical aperture and also extends the trap's focus along thez axis. The trap's effective numerical aperture in water would beroughly 1.2. The effect of spherical aberration can be approximatelycorrected by pre-distorting the beam of light 20 with the additionalphase profile,

$\begin{matrix}{{{\phi_{a}(\rho)} = {\frac{a}{\sqrt{2}}\left( {{6x^{4}} - {6x^{2}} + 1} \right)}},} & (4)\end{matrix}$

the Zernike polynomial describing spherical aberration. The radius, x,is measured as a fraction of the optical train aperture, and thecoefficient α is measured in wavelengths of light. This procedure isused to correct for small amount of aberration present in practicaloptical trapping systems to optimize their performance.

This correction was applied to an array 110 of 35 optical tweezers shownas a three-dimensional reconstruction in FIG. 3E. These optical traps100 are arranged in a three-dimensional body-centered cubic (BCC)lattice 115 shown in FIG. 3F with a 10.8 μm lattice constant. Withoutcorrecting for spherical aberration, these traps 100 would blend intoeach other along the optical axis 90. With correction, their axialintensity gradients are clearly resolved. This accounts for holographictraps' ability to organize objects along the optical axis.

Correcting for aberrations reduces the range of displacements, z, thatcan be imaged. Combining φ_(α)(ρ) with φ_(z)(ρ) and φ₀(ρ) increasesgradients in φ(ρ), particularly for larger values of ρ near the edges ofthe diffraction optical element. Diffraction efficiency falls offrapidly when |Vφ(ρ)| exceeds 2π/Δρ, the maximum phase gradient that canbe encoded on the SLM 60 with pixel size Δρ. This problem is exacerbatedwhen φ₀(ρ) itself has large gradients. In a preferred embodiment morecomplex trapping patterns without aberration are prepared. Inparticular, we use uncorrected volumetric imaging to illustrate thecomparative advantages of the extended optical traps 100.

The extended optical traps 100 have been projected in a time-sharedsense by rapidly scanning a conventional optical tweezer along thetrap's intended contour. A scanned trap has optical characteristics asgood as a point-like optical tweezer, and an effective potential energywell that can be tailored by adjusting the instantaneous scanning rateKinematic effects due to the trap's motion can be minimized by scanningrapidly enough. For some applications, however, continuous illuminationor the simplicity of an optical train with no scanning capabilities canbe desirable.

Continuously illuminated line traps have been created by expanding anoptical tweezer 125 along one direction (see FIG. 4A). This can beachieved, for example, by introducing a cylindrical lens component suchas by element 130 (see FIG. 1) into the objective's input plane.Equivalently, a cylindrical-lens line tweezer can be implemented byencoding the function φ_(c)(ρ)=πz₀ρ_(x) ²/(λƒ²) on the SLM 60. Theresult, shown in FIGS. 4A-4D appears best useful in the plane of bestfocus, z=z₀, with the point-like tweezer having been extended to a linewith nearly parabolic intensity and a nearly Gaussian phase profile. Thethree-dimensional reconstruction, however, reveals that the cylindricallens component merely introduces a large amount of astigmatism into thebeam of light 20, creating a second focal line perpendicular to thefirst. This is problematic for some applications because the astigmaticbeam's axial intensity gradients are far weaker than a conventionaloptical tweezer's. Consequently, cylindrical-lens line traps typicallycannot localize objects against radiation pressure along the opticalaxis 90.

Replacing the single cylindrical lens with a cylindrical Kepleriantelescope for the element 130 eliminates the astigmatism and thuscreates a stable three-dimensional optical trap. Similarly, using theobjective lens 50 to focus two interfering beams creates aninterferometric optical trap capable of three-dimensional trapping.These approaches, however, offer little control over the extended trapsintensity profiles, and neither affords control over the phase profile.

Shape-phase holography provides absolute control over both the amplitudeand phase profiles of an extended form of the optical trap 100 at theexpense of diffraction efficiency. It also yields traps with optimizedaxial intensity gradients, suitable for three-dimensional trapping. Ifthe line trap is characterized by an amplitude profile ũ(ρ_(x)) and aphase profile {tilde over (p)}(ρ_(x)) along the {circumflex over(ρ)}_(x) direction in the objective's focal plane, then the field in theSLM plane is given from Eq. (1) as,

ψ(ρ)=u(ρ_(x))exp(ip(ρ_(x))),  (5)

where the phase p(ρ_(x)) is adjusted so that u(ρ_(x))≧0. Shape-phaseholography implements this one-dimensional complex wavefront profile asa two-dimensional phase-only hologram,

$\begin{matrix}{{\phi (\rho)} = \left\{ \begin{matrix}{{p\left( \rho_{x} \right)},} & {{S(\rho)} = 1} \\{{q(\rho)},} & {{{S(\rho)} = 0},}\end{matrix} \right.} & (6)\end{matrix}$

where the shape function S(ρ) allocates a number of pixels along the rowρ_(y) proportional to u(ρ_(x)). One particularly effective choice is forS(ρ) to select pixels randomly along each row in the appropriaterelative numbers. The unassigned pixels then are given values q(ρ) thatredirect the excess light away from the intended line. Typical resultsare presented in FIG. 5A.

Unlike the cylindrical-lens trap, the holographic line trap 130 in FIGS.5A-5D focuses as a conical wedge to a single diffraction-limited line inthe objective's focal plane. Consequently, its transverse angle ofconvergence is comparable to that of an optimized point trap. This meansthat the holographic line trap 120 has comparably strong axial intensitygradients, which explains its ability to trap objects stably againstradiation pressure in the z direction.

The line trap's transverse convergence does not depend strongly on thechoice of intensity profile along the line. Its three-dimensionalintensity distribution, however, is very sensitive to the phase profilealong the line. Abrupt phase changes cause intensity fluctuationsthrough Gibbs phenomenon. Smoother variations do not affect theintensity profile along the line, but can substantially restructure thebeam. The line trap 120 created by the cylindrical lens element 130 forexample, has a parabolic phase profile. Inserting this choice into Eq.(2) and calculating the associated shape-phase hologram with Eqs. (1)and (6) yields the same cylindrical lens phase profile. This observationopens the door to applications in which the phase profile along a linecan be tuned to create a desired three-dimensional intensitydistribution, or in which the measured three-dimensional intensitydistribution can be used to assess the phase profile along the line.These applications will be discussed elsewhere.

The foregoing description of embodiments of the present invention havebeen presented for purposes of illustration and description. It is notintended to be exhaustive or to limit the present invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of thepresent invention. The embodiments were chosen and described in order toexplain the principles of the present invention and its practicalapplication to enable one skilled in the art to utilize the presentinvention in various embodiments, and with various modifications, as aresuited to the particular use contemplated.

1. A method of obtaining a cross-section of an optical trap useful forreconstruction and characterization of the optical trap, comprising thesteps of: providing an optical train; providing a beam of light to theoptical train; applying a predetermined hologram to the beam of light toform an optical trap; further applying a phase only hologram toestablish a predetermined optical trap configuration; and applying theoptical train to carry out at least one of reconstruction andcharacterization of the optical trap configuration.
 2. The method asdefined in claim 1 further including the steps of using the optical trapto manipulate an object in the same plane.
 3. The method as defined inclaim 1 wherein the phase only hologram is applied to selected ones of aplurality of the optical trap, the step including modifying wavefrontsof the optical trap.
 4. The method as defined in claim 1 wherein thesteps of applying a phase only hologram to carry out at least one ofreconstruction and characterization of the optical trap includes havingthe optical train with a mirror disposed in the sample plane andoperating on the beam of light to process a light field of the opticaltrap.
 5. The method as defined in claim 4 wherein the mirror disposed inthe sample plane is translated relative to an objective lens of theoptical train to perform a reconstruction or characterization ofintensity of the optical trap.
 6. The method as defined in claim 4wherein the optical trap is translated relative to a fixed form of themirror by including in the hologram a parabolic phase function.
 7. Themethod as defined in claim 6 wherein the parabolic phase functioncomprises,${{\phi_{z}(\rho)} = {- \frac{{\pi\rho}^{2}z}{\lambda \; f^{2}}}},$where, φ=distance or position within the hologram; z=distance along anoptical axis of the optical train; λ=wavelength of the light;ƒ=objective lens focal length.
 8. The method as defined in claim 6wherein computer software executed by a computer creates the hologramfor translating the optical trap.
 9. The method as defined in claim 2further including the step of passing the beam of light through anobjective lens of the optical train and applying a phase profile to theoptical trap, thereby enabling the plurality of optical traps toorganize selected ones of the objects along an optical axis of theoptical train.
 10. The method as defined in claim 9 wherein the phaseprofile comprises,${{\phi_{a}(\rho)} = {\frac{a}{\sqrt{2}}\left( {{6x^{4}} - {6x^{2}} + 1} \right)}},$thereby correcting for spherical aberration where x is distance in aplane perpendicular to the optical axis.
 11. The method as defined inclaim 2 further including the step of forming a single optical trap andrapidly scanning the optical trap along a predetermined optical trapconfiguration to enable manipulation of the object.
 12. The method asdefined in claim 1 further including the step of introducing anobjective lens in the optical train and introducing a cylindrical lenscomponent into an input plane of the objective lens.
 13. The method asdefined in claim 1 wherein the step of applying a predetermined hologramincludes introducing a cylindrical-lens line tweezer component into thehologram.
 14. The method as defined in claim 13 wherein thepredetermined hologram includes the functionφ_(c)(ρ)=φ_(c)(ρ)=πz_(o)ρ_(x) ²/(λƒ²).
 15. The method as defined inclaim 1 further including in the optical train a Keplerian telescope foreliminating astigmatism, thereby creating a stable three dimensionaloptical trap.
 16. The method as defined in claim 1 wherein the opticaltrap comprises a hologram constructed of, a. an amplitude profileũ(ρ_(x)), b. a phase profile {tilde over (p)}(ρ_(x)) with an objectivefocal plane direction {circumflex over (ρ)}_(x).
 17. The method asdefined in claim 14 wherein the optical train includes an SLM having anassociated plane and the field in the associated plane is given by,ψ(ρ)=u(ρ_(x))exp(ip(ρ_(x))).
 18. The method as defined in claim 1wherein the phase only hologram comprises,${\phi (\rho)} = \left\{ \begin{matrix}{{p\left( \rho_{x} \right)},} & {{S(\rho)} = 1} \\{{q(\rho)},} & {{S(\rho)} = 0.}\end{matrix} \right.$
 19. The method as defined in claim 1 wherein theat least one of reconstruction and characterization includesestablishing a two dimensional cross section of original trappingdensity of the predetermined optical trap configuration.
 20. The methodas defined in claim 19 wherein a three dimensional reconstruction orcharacterization of the optical trap configuration is obtained from aplurality of the two dimensional cross sections.
 21. A system forobtaining a cross-section of an optical trap useful for reconstructionand characterization of the optical trap, comprising: an optical train;a source of light to produce a beam of light with the light input to theoptical train; a spatial light modulator; and a computer having acomputer program to control the spatial light modulator to generate aphase controlled hologram, the hologram providing a predeterminedconfiguration of optical traps; and a mirror in the optical train anddisposed in the sample plane for operating on the beam of light to atleast one of reconstruct and characterize a light field of each of theoptical traps;
 22. The system as defined in claim 21 wherein thehologram comprises a phase only hologram.
 23. The system as defined inclaim 21 further including at least one of (1) the mirror beingtranslated along the optical train relative to the objective lens, toperform at least one of a three dimensional reconstruction, volumetricimaging and characterization of the light field of each of the opticaltraps and (2) a modified parabolic phase function hologram being appliedto the phase controlled hologram to translate the optical traps relativeto a fixed form of the mirror.
 24. A method of obtaining a cross-sectionof the trapping intensity of trapping light, comprising the steps of:providing an optical train; providing a beam of light to the opticaltrain; applying a predetermined hologram to the beam of light togenerate trapping light; reflecting the trapping light off a mirror; andobtaining a cross-section of trapping intensity of the trapping lightfrom the reflected light.
 25. The method as defined in claim 24 whereinthe trapping light is reflected off a front surface mirror in a sampleplane.
 26. The method as defined in claim 24 wherein the trapping lightis reflected off a front surface mirror into an objective lens
 27. Themethod as defined in claim 24 wherein the trapping light is reflectedoff a front surface mirror into an objective lens and further transmitsimages of the trapping light through a partially reflecting mirror. 28.The method as defined in claim 24 wherein the trapping light isreflected off a front surface mirror into an objective lens and furthertransmits images of the trapping light through a partially reflectingmirror into a camera.
 29. The method as defined in claim 24 wherein thetrapping light is translated relative to the mirror
 30. The method asdefined in claim 24 wherein the trapping light is translated relative tothe mirror by superimposing a phase function.
 31. The method as definedin claim 30 wherein the phase function comprises a parabolic phasefunction.
 32. The method as defined in claim 26 wherein an image isconstructed from the reflected light which represents a cross-section ofthe trapping intensity at a particular distance from a focal plane ofthe objective lens.
 33. The method as defined in claim 24 wherein thecross-section is reconstructed at a plurality of distances z from afocal plane of the optical train.
 34. The method as defined in claim 24wherein a plurality of images are used to create a volumetricrepresentation of the trapping intensity.
 35. The method as defined inclaim 28 wherein an image is constructed from the light reflected of thefront surface mirror and which represents the cross-section of thetrapping intensity at a particular distance from a focal plane of theobjective lens and wherein the cross section is reconstructed at aplurality of distances from the focal plane and are used to create avolumetric representation of the trapping intensity.
 36. The method asdefined in claim 35 wherein the trapping light is translated relative tothe mirror by superimposing a phase function.
 37. A system for obtaininga cross-section of the trapping intensity of trapping light, comprising:an optical train; a source for providing a beam of light to the opticaltrain; a computer for applying a predetermined hologram to the beam oflight to generate trapping light; a mirror for reflecting the trappinglight; and a sensor for obtaining a cross-section of trapping intensityof the trapping light from the reflected light thereby enabling at leastone of analysis of the trapping intensity and use of the cross-sectionfor operating on a sample.
 38. The system as defined in claim 37 furtherincluding a front surface form of the mirror with the trapping lightreflected off the front surface mirror into an objective lens andfurther transmits images of the trapping light through a partiallyreflecting mirror into a camera.
 39. The system as defined in claim 37wherein the trapping light is translated relative to the mirror by thecomputer superimposing a phase function on the trapping light.
 40. Thesystem as defined in claim 37 wherein an image is constructed from thereflected light which represents a cross-section of the trappingintensity reconstructed at least one of (1) at a particularreconstructed distance from a focal plane of the objective lens; (2) aplurality of distances z from a focal plane of the optical train; and(3) with a plurality of the images being used to create a volumetricrepresentation of the trapping intensity.